| L(s) = 1 | − 2·2-s − 9.74·3-s + 4·4-s + 19.4·6-s − 8·8-s + 67.9·9-s + 6.52·11-s − 38.9·12-s + 41.6·13-s + 16·16-s + 109.·17-s − 135.·18-s + 29.7·19-s − 13.0·22-s − 180.·23-s + 77.9·24-s − 83.3·26-s − 399.·27-s + 183.·29-s − 116.·31-s − 32·32-s − 63.5·33-s − 219.·34-s + 271.·36-s − 396.·37-s − 59.4·38-s − 406.·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s + 1.32·6-s − 0.353·8-s + 2.51·9-s + 0.178·11-s − 0.937·12-s + 0.888·13-s + 0.250·16-s + 1.56·17-s − 1.78·18-s + 0.359·19-s − 0.126·22-s − 1.63·23-s + 0.663·24-s − 0.628·26-s − 2.84·27-s + 1.17·29-s − 0.675·31-s − 0.176·32-s − 0.335·33-s − 1.10·34-s + 1.25·36-s − 1.76·37-s − 0.253·38-s − 1.66·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 9.74T + 27T^{2} \) |
| 11 | \( 1 - 6.52T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 183.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 396.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 405.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 67.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 510.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 352.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 59.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 133.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 571.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 783.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.046072593768228513777666897748, −7.39212498603648702241823787062, −6.53041164731191933014231701056, −5.91709804566221791616332816740, −5.40290426981459775394609081384, −4.34124628354409738060968156188, −3.38749995983044838676946770631, −1.68273410271621471651294269805, −0.992499911325811256428620954577, 0,
0.992499911325811256428620954577, 1.68273410271621471651294269805, 3.38749995983044838676946770631, 4.34124628354409738060968156188, 5.40290426981459775394609081384, 5.91709804566221791616332816740, 6.53041164731191933014231701056, 7.39212498603648702241823787062, 8.046072593768228513777666897748
